Quarkonia and Quark Drip Lines in Quark-Gluon Plasma

Quarkonia and Quark Drip Lines in Quark-Gluon Plasma Cheuk-Yin Wong

arXiv:hep-ph/0606200v2 19 Jun 2007

Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 and Department of Physics, University of Tennessee, Knoxville, TN 37996 (Dated: February 2, 2008) ¯ potential by using the thermodynamic quantities obtained in lattice gauge We extract the Q-Q calculations. The potential is tested and found to give spontaneous dissociation temperatures that ¯ potential, we agree well with those from lattice gauge spectral function analysis. Using such a Q-Q examine the quarkonium states in a quark-gluon plasma and determine the ‘quark drip lines’ which ¯ states from the unbound region. The characteristics separate the region of bound color-singlet QQ ¯ states with light quarks to of the quark drip lines severely limit the region of possible bound QQ temperatures close to the phase transition temperature. Bound quarkonia with light quarks may exist very near the phase transition temperature if their effective quark mass is of the order of 300-400 MeV and higher. PACS numbers: 25.75.-q 25.75.Dw

I.

INTRODUCTION

The degree to which the constituents of a quark-gluon plasma (QGP) can combine to form composite entities is an important property of the plasma. It has significant implications on the nature of the phase transition, the quark-gluon plasma equation of state, the probability of recombination of plasma constituents prior to the phase transition, and the chemical yields of the observed bound hadrons. The successes of the recombination model [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] suggests that quark partons may form bound or quasi-bound states in the quark-gluon plasma, at least at, or close to, the phase transition temperature. It is an important theoretical question as to the range of temperatures in which these quarkonia may be bound or quasi-bound. The successes of the thermal model [1, 2, 3, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] for hadron yields also raises the important question whether hadrons may become bound or quasi-bound in the quark-gluon plasma. If they are indeed bound in the quark-gluon plasma, the approach to chemical equilibrium may commence in the quark-gluon plasma phase before the phase transition and the boundaries of the quark-gluon plasma phase and the hadron phase may overlap. Recent spectral analyses of quarkonium correlators indicated that J/ψ may be stable up to 1.6Tc where Tc is the phase transition temperature [21, 22]. Subsequently, there has been renewed interest in quarkonium states in ¯ states with light quarks may be bound up to a few Tc quark-gluon plasma as Zahed and Shuryak suggested that QQ [23]. Quarkonium bound states and instanton molecules in the quark-gluon plasma have been considered by Brown, Lee, Rho, and Shuryak [24]. As heavy quarkonia may be used as a diagnostic tool [25], there have been many recent investigations on the stability of heavy quarkonia in the plasma [26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Previously, DeTar [40], Hansson, Lee, and Zahed [41], and Simonov [42] observed that the range of strong interaction is not likely to change drastically across the phase transition and suggested the possible existence of relatively narrow ¯ states in the plasma. On the other hand, Hatsuda and Kunihiro [43] considered the persistence of low-lying QQ soft modes in the plasma which may manifest themselves as pion-like and sigma-like states. The use of baryonstrangeness correlation and charge fluctuation to study the abundance of light quarkonium states in the plasma have been suggested recently [44, 45]. We would like to investigate the composite properties of the plasma and to determine its ‘quark drip lines’. We shall ¯ states and the Q-Q ¯ quantities in this paper refer to those of color-singlet Q-Q ¯ focus our attention on color-singlet Q-Q states unless specified otherwise. Here we follow Werner and Wheeler [46] and use the term ‘drip line’ to separate ¯ states from the unbound region of spontaneous quarkonium dissociation. It the region of bound color-singlet QQ should be emphasized that a quarkonium can be dissociated by collision with constituent particles to lead to the corresponding ‘particle-dissociation lines’, which can be interesting subjects for future investigation. We would like to use the potential model to study the stability of quarkonia, as the potential model can be used to evaluate many more quantities than the lattice gauge spectral function analysis. The potential model lends itself to extrapolation into unknown regions of quark masses and temperatures. An important physical quantity in the ¯ potential between the quark Q and the antiquark Q ¯ at a separation R at a temperature T . potential model is the Q-Q Previous work in the potential model uses the color-singlet free energy F1 (R, T ) [26, 31, 38] or the color-singlet internal ¯ potential, without rigorous theoretical energy U1 (R, T ) [23, 32, 37] obtained in lattice gauge calculations as the Q-Q ¯ system. justifications. Here, the subscripts of U1 (R, T ) and F1 (R, T ) refer to the color-singlet property of the Q-Q The internal energy U1 (R, T ) is significantly deeper and spatially more extended than the free energy F1 (R, T ). The degree of quarkonium binding will be significantly different whether one uses the internal energy U1 (R, T ) or the

2 ¯ potential. Treating the internal energy U1 (R, T ) as the Q-Q ¯ potential led Shuryak free energy F1 (R, T ) as the Q-Q and Zahed to suggest the possibility of color-singlet quarkonium states with light quarks in the plasma [23]. The ¯ potential. conclusions will be quite different if one uses the free energy F1 (R, T ) as the Q-Q ¯ While F1 (R, T ) or U1 (R, T ) can both be used as the Q-Q potential at T = 0 (at which F1 (R, T ) = U1 (R, T )), the situation is not so clear in a thermalized quark-gluon plasma. It is important to find out the meaning of these ther¯ potential. modynamical quantities calculated in the finite-temperature lattice gauge theory so as to extract the Q-Q ¯ in a thermal medium, the Q-Q ¯ potential in the If one constructs the Schr¨odinger equation for the Q and Q ¯ when the medium particles have re-arranged Schr¨odinger equation contains those interactions that act on Q and Q, themselves self-consistently. On the other hand, the total internal energy U1 (R, T ) contains not only these interactions ¯ but also the internal gluon energy Ug (R, T ) relative to the gluon internal energy Ug0 in the that act on Q and Q, ¯ absence of Q and Q, as shown deductively in Ref. [27] starting from the definition of the free energy in lattice gauge theory in quenched QCD. If the gluon internal energy Ug (R, T ) were independent of R, then U1 (R) could well be ¯ potential. However, in the grand canonical ensemble, Ug (R, T ) depends on R. To get the Q-Q ¯ used as the Q-Q potential, it is therefore necessary to subtract Ug (R, T ) − Ug0 from U1 (R, T ). As the subtleties of these results may not appear evident and the problem of non-perturbative QCD so intrinsically complicated, a thorough understanding of an analogous, but not identical, problem in QED is worth having. Therefore, we examine in detail the simple QED ¯ in a massless charged medium in a grand canonical ensemble, where case of Debye screening of charges Q and Q ¯ the results can be readily obtained analytically. We would like to show that there is a relationship between the Q-Q ¯ potential between two static potential and the total internal energy when screening occurs: the Debye screening Q-Q opposite charges in QED can be obtained from the total internal energy by subtracting out the internal energy of the medium particles. The results in the Debye screening case in QED support our previous conclusion in Ref. [27] that in the QCD lattice gauge calculations in the grand canonical ensemble, it is necessary to subtract out the R-dependent internal ¯ in the plasma. energy of the QGP from the total internal energy in order to obtain the potential between Q and Q ¯ Additional lattice gauge calculations may be needed to evaluate the QGP internal energy in the presence of Q and Q. It is nonetheless useful at this stage to suggest approximate ways to evaluate the QGP internal energy. We proposed earlier a method by making use of the equation of state of the quark-gluon plasma obtained in an independent lattice gauge calculation [27]. The equation of state provides a relationship between the QGP internal energy and the QGP ¯ potential can be represented as entropy content. As the QGP entropy content is the difference U1 − F1 , the Q-Q a linear combination of U1 and F1 , with coefficients depending on the quark-gluon plasma equation of state. The proposed potential was tested and found to give spontaneous dissociation temperatures that agree well with those from lattice gauge spectral function analysis in the quenched approximation. The comparison of the potential model results with those from spectral analyses in the same quenched approximation is useful as a theoretical test of the potential model. However, in the quenched approximation, the quark-gluon plasma is assumed to consist of gluons only and the effects of dynamical quarks have not been included. As dynamical quarks provide additional screening, one wishes to know whether this additional screening will modify the binding energies of quarkonia significantly or not. The presence of dynamical quarks also lowers the phase transition temperature from 269 MeV for quenched QCD to 154 MeV for full QCD with three flavors [47]. For these reasons, it is necessary to include dynamical quarks to assess their effects on the stability of quarkonia. The knowledge of the single-particle states using potentials extracted from lattice gauge calculations in full QCD can then be used to examine the stability of both heavy and light quarkonia and to determine the location of the quark drip lines. We focus our attention mainly on heavy quarkonia for which a non-relativistic treatment is a good description. However, the problem of the stability of quarkonia with light quarks is intrinsically so complicated and the question of their stability up to a temperature of few units of Tc so important [1]-[23] that even an approximate estimate using the non-relativistic potential model is worth having. The subject of light quarkonia will be examined again, with the inclusion of the relativistic effects as in recent works [48, 49], in the course of time. The authors of Refs. [50, 51] claim recently that potential models cannot describe heavy quarkonia above Tc , as their potential model correlators fail to reproduce lattice gauge correlators for all cases with all types of potentials. Such a complete disagreement for all cases and all types of potentials suggests that the lack of agreement may not be due to the potential model (or models) themselves but to their method of evaluating the meson correlators in the potential model. We show recently that when the contributions from the bound states and continuum states are properly treated [30], the potential model correlators obtained with the proposed potential in Ref. [27] are consistent with lattice gauge correlators. This paper is organized as follows. In Section II, we describe the puzzling behavior of the increase of the entropy of ¯ increases. To understand such a behavior, we introduce in Section the QCD medium as the separation between Q and Q III a simple model of Debye screening in QED for which various thermodynamic quantities can be readily calculated. ¯ pair in Debye In Section VI, the variation of the number density and entropy density of the medium particles for a Q-Q ¯ separation R when the second-order contributions are included. In Section screening is shown to depend on the Q-Q

3 V, we find similarly that the internal energy of the medium particles in Debye screening also increases with R when second-order contributions are included. In Section VI, we examine the Schr¨odinger equation for the relative motion ¯ and identify the potential between Q and Q ¯ in Debye screening. We reach the conclusion that in order of Q and Q to obtain the Debye screening potential between two static charges in the grand canonical ensemble, it is necessary to subtract out the internal energy of the medium particles from the total internal energy. Returning to lattice gauge calculations in Section VII, we suggest a method in which the medium internal energy can be approximately ¯ determined and subtracted by using the information on the quark-gluon plasma of state. Consequently, the Q-Q potential turns out to be a linear combination of U1 (R, T ) and F1 (R, T ), with coefficients depending on the equation of state. In Section VIII, we use different potentials to calculate the spontaneous dissociation temperatures for various quarkonia and compare them with those obtained from lattice gauge spectral function analyses in quenched QCD. The dissociation temperatures obtained with the proposed linear combination of U1 and F1 give the best agreement with those from the spectral function analyses. In Section IX, we show how the thermodynamical quantities in full QCD with two flavors are parametrized. The dissociation temperatures for various heavy quarkonia for full QCD with two flavors are obtained in Section X. We introduce the quark drip lines in quark-gluon plasma in Section XI. We conclude and summarize our discussions in Section XII. II.

THERMODYNAMICAL QUANTITIES IN LATTICE GAUGE CALCULATIONS AND DEBYE SCREENING

Thermodynamical quantities for a heavy quark pair in the color-singlet state was studied by Kaczmarek and Zantow in quenched QCD and in full QCD with two flavors [33, 34]. They calculated htrL(r/2)L† (−R/2)i and obtained the color-singlet free energy F1 (R, T ) from htrL(R/2)L† (−R/2)i = e−F1 (R,T )/kT .

(1)

Here trL(R/2)L† (−R/2) is the trace of the product of two Polyakov lines at R/2 and −R/2. The free energy ¯ pair, is measured relative to the free energy without the Q-Q ¯ pair. The quark F1 (R, T ), in the presence of the Q-Q and the antiquark lines do not, in general, form a close loop. As a gauge transformation introduces phase factors at the beginning and the end of an open Polyakov line, htrL(R/2)L† (−R/2)i is not gauge invariant under a gauge transformation. Calculations have been carried out in the Coulomb gauge which is the proper gauge to study bound states. From the free energy F1 , Kaczmarek and Zantow [33, 34] calculated the internal energy U1 using the statistical identity U1 (R, T ) = F1 (R, T ) + T S1 (R, T ),

(2)

¯ pair and is where S1 (R, T ) = −∂F1 (R, T )/∂T is the entropy of the system in the presence of a color-singlet Q-Q ¯ measured relative to the entropy of the system in the absence of the Q-Q pair. ¯ potential from thermodynamical quantities calculated in the lattice gauge theory, we In order to extract the Q-Q need to understand the behavior of the free energy F1 (R, T ), the internal energy U1 (R, T ), and T times the entropy, T S1 (R, T ), which we shall also abbreviatingly call “the entropy” for simplicity of nomenclature. As these three quantities are related by Eq. (2) we need to find out the behavior of only two of these three quantities. Following the terminology used in lattice gauge calculations [34], U1 , F1 , and T S1 are defined as being measured relative to their ¯ corresponding quantities in the absence of Q and Q. We can begin by studying the entropy T S1 (R, T ) of the system. We note that the lattice gauge calculations show that the total entropy T S1 (R) increases as a function of R, and saturates after the separation reaches a large value of R [34] as shown in Fig. 1(a). What does such a behavior tell us about the response of the medium particles to the ¯ in a thermal bath? presence of the external color sources of Q and Q ¯ and the quark-gluon plasma. For In the system under consideration, the system consists of the Q, the antiquark Q simplicity we can examine the quenched case for which the quark-gluon plasma is assumed to consist of gluons only. ¯ and the gluons. However, in the The entropy of the system therefore comes from the sum of the entropies of Q, Q ¯ lattice gauge calculations in a thermal bath, the Q and the Q are held fixed and do not contribute to the entropy of the system. The entropy of the system T S1 comes entirely from the gluons. By fixing a temperature and focusing our attention at the state of thermal equilibrium in lattice gauge calculations, the gluons are in a grand canonical ensemble in contact with the thermal bath. The content and thermodynamical properties of the gluons are determined by the condition of thermal equilibrium at the fixed temperature of the thermal bath. The observed behavior of the entropy T S1 (R) as a function of R in Fig. 1(a) suggests that the gluon entropy content increases as the separation R increases, and the entropy saturates when R reaches a certain limit.

4

TS1 (MeV)

500 400 300

lattice gauge results, T=1.3Tc

100 0 0.0

TS(R) / TS(R→∞)

(a)

200

0.5

1.5

1.0 R (fm)

1 0.8 0.6 0.4

(b)

0.2 0 0.0

1.0

2.0 R / rDebye

3.0

4.0

FIG. 1: (a) The total entropy T S1 (R, T ) as a function of R at T = 1.3Tc from the lattice gauge calculations [34]. (b) The ratio η(R) = T S(R)/T S(R → ∞) (which is also equal to N (R)/N (R → ∞) and U (R)/U (R → ∞)) as a function of R/rD in a thermal medium under Debye screening.

As we know from the work of Landau and Belenkii [52], the entropy content of a system is closely correlated with the number content of the particles in the system. The behavior of the entropy suggests that the number of gluons increases as the separation R increases. How do we understand such a behavior? If the number of gluons increases as a function of R, what happens to the internal energy content of the gluons as a function of R, and how does that ¯ we wish to extract? affect the total internal energy and the potential between Q and Q III.

ANALOGOUS PROBLEM OF DEBYE SCREENING IN QED

The QCD problem of a quark and an antiquark in the presence of the actions of the gluons is so complicated that it is worth having even a good understanding of the puzzling behavior in an analogous, but not identical, problem ¯ under the action of charged medium particles in QED. of a static positive charge Q and another negative charge Q We would like to ask whether there is a similar behavior of the number and the entropy of medium particles as the ¯ increases. If the number of medium particles separation R between the positive charge Q and the negative charge Q increases as a function of R, what happens to the internal energy content of the system and the medium particles as ¯ and total internal energy we a function of R, and what is the relationship between the potential between Q and Q wish to extract? Accordingly, we study the simple system of Debye screening and consider a Q with charge +q held fixed at −R/2 ¯ with a charge −q at R/2, in a medium of massless fermions with charges e± = ±q, in a thermal bath of and a Q temperature T in a grand canonical ensemble. Any pair of charged particles with charges e1 and e2 separated by a distance r are assumed to interact with a Coulomb interaction e1 e2 /|r|. To make the problem simple, we shall assume the attainment of local thermal equilibrium. ¯ are introduced into the medium, the medium fermions will re-arrange themselves When the external charges Q and Q in both coordinates and momenta to reach a new local thermal equilibrium. The total number of the medium particles in the system is not a constant of motion but is determined by the condition of thermal equilibrium, maintained by the thermal bath. ¯ are separated by a distance R at a temperature T , the self-consistent reWhen the external charges Q and Q arrangement of the medium charged fermion particles leads to a potential V (r, R) at a point r and under a local thermal equilibrium, the momentum distribution of the medium particles at r in the Born-Oppenheimer treatment is

5 given by f± (r, p, R) =

1 . exp{[p + e± V (r, R)]/T } + 1

(3)

Here and henceforth, the + and − subscripts designate quantities for the positive and negative medium particles respectively. From the above Wigner function distribution, we can obtain various thermodynamic quantities. The integration ¯ of the Wigner function over all momenta gives the spatial number density distribution n± (r, R) at r, when Q and Q are separated by R, Z g p2 dp f± (r, p, R), (4) n± (r, R) = 2 2π where g is the degeneracy of the levels. We can consider the high temperature case for which it is useful to expand various quantities as a power series of e± V ((r, R)/T . Up to the second order in [e± V (r, R)/T ]2 , the medium particle density is  (5) n± (r, R) = n0± 1 − a1 [e± V (r, R)/T ] + a2 [e± V (r, R)/T ]2 ,

where

n0± =

a1 =

gT 3 3 ζ(3)Γ(3), 2π 2 4

1 2 ζ(2) 3 4 ζ(3)

(6)

= 0.91233,

(7)

= 0.3845.

(8)

and a2 =

1 2 ζ(1) 3 4 ζ(3)

In passing, we note that if the medium particles obeys Boltzmann statistics, the coefficients a1 and a2 would equal a1 = 1 and a2 = 0.5, as it follows from the expansion of the well-known Boltzmann distribution n± (r, R) = n0± exp{−e±V (r, R)/T } for Boltzmann particles in an external field. The values of the a1 and a2 coefficients for Boltzmann statistics differ only slightly from the corresponding values in (7) and (8) for fermion particles. ¯ are separated by R, is given by For the Fermi-Dirac medium particles, the entropy density at r, when Q and Q Z g σ± (r, R) = 2 p2 dp {−f± ln f± − (1 − f± ) ln(1 − f± )} . (9) 2π Upon substituting the Fermi-Dirac distribution of Eq. (3) into the above equation, we find    Z 4p g 2 + e± V (r, R) T σ± (r, R) = p dpf± (r, p, R) 2π 2 3 4p ≡ h + e± V (r, R)i / T, 3

(10)

and we obtain

where

 0 1 − b1 [e± V ((r, R)/T ] + b2 [e± V (r, R)/T ]2 , σ± (r, R) = σ± 0 σ± =

b1 =

gT 3 7 4 ζ(4) Γ(4), 2π 2 8 3

3 4 4 ζ(3)[ 3 Γ(4) − Γ(3)] 4 7 8 ζ(4) 3 Γ(4)

= 0.7139,

(11)

(12)

(13)

6 and b2 =

4 Γ(4) 1 2 ζ(2)[ 3 2 − Γ(3)] 7 4 8 ζ(4) 3 Γ(4)

= 0.2171.

(14)

Here we have purposely written the numerators of the b1 and b2 coefficients as a difference where the first term comes from h4p/3i and the second term comes from he± V i of Eq. (10). The evaluation of various thermodynamical quantities requires the knowledge of V (r, R). To determine V (r, R) ¯ are separated by R, self-consistently, we have the charge density at the point r, when Q and Q ρtotal (r, R) = qδ(r +

R R ) − qδ(r − ) + e+ n+ (r, R) + e− n− (r, R). 2 2

(15)

Using the number density distribution given in Eq. (5), the charge density, up to the second power in e± V (r, R)/T , becomes ρtotal (r, R) = qδ(r +

R R ) − qδ(r − ) − n0 a1 q 2 V (r, R), 2 2

(16)

where n0 = n0+ + n0− and the zeroth-order and second-order terms of Eq. (5) cancel out on account of the presumed charge neutrality of the system for which n0+ = n0− . The Poisson equation for the potential is then given by   R R 2 2 ∇r V (r, R) = −4π q δ(r + ) − q δ(r − ) − n0 a1 q V (r, R) , (17) 2 2 which has the solution qe−µr− qe−µr+ − , r+ r− r± = |r ± R/2|, r 4πn0 a1 q 2 1 µ = , = T rD

V (r, R) =

(18a) (18b) (18c)

where µ is the Debye mass and rD is the Debye screening length. IV.

VARIATION OF NUMBER DENSITY AND ENTROPY DENSITY WITH R IN DEBYE SCREENING

The simple solution V (r, R) in Eq. (18) of the last section allows us to have a profile of the self-consistent medium particle number density and entropy density in all spatial points at local thermal equilibrium at T . In Eqs. (5) and (11), the coefficients of a1 , a2 , b1 and b2 are all positive. For positive medium particles, the first-order increment in the number density n+ (r, R) and σ+ (r, R), [also u+ (r, R) in Eq. (32)] are therefore measured by [−e+ V (r, R)/T ] illustrated in Fig. 2 as a function of ρ/rD and z/rD . As one observes, the first-order contributions, given by −e+ V (r, R)/T , represent a depletion for the positive medium particles near the positive static charge Q at −R/2, and an enhance¯ at R/2. The degree of depletion and the degree of enhancement are equal and ment near the negative static charge Q opposite to each other. When we sum over all spatial points, the sum of the first-order depletion and enhancement cancel each other to give a zero total contribution. The second-order contributions are proportional to [e+ V (r, R)/T ]2 and are always positive. They are illustrated ¯ separations. They always enhance the number density and in Fig. 3 as a function of ρ/rD and z/rD for different Q-Q the entropy density. The enhancement is small when the two static charges are close together in Fig. 3(a), as there is a substantial cancellation of the two terms in Eq. (18a). The enhancement reaches a constant value when the static ¯ reaches a separation of 1-2 units of the Debye screening length as shown in Fig. 3b and 3c. charges Q and Q If one integrates over all spatial points to R obtain the total number of positive charge medium particles, one finds that the integration over the first-order term, dr[−e+ V (r, R)/T ], is zero because the depletion cancels the enhancement. However, the second-order contributions always give a positive contribution, and the number of positive medium ¯ is given by particles, measured relative to its corresponding quantity in the absence of Q and Q, Z Z  (19) N+ (R) = dr n+ (r, R) − n0+ = n0+ dr a2 [e+ V (r, R)/T ]2 .

7 -2 0.8 0.6 0.4 0.2

=rD

e+ V =T q 2 =rD T

z=rD

-2 0 0.8 0.6 1 0.4 0.2

-1

-1

z=rD 0

2

-2 0.8 1 0.6 2 0.4 0.2

0

0

0

4

4

4

2

2

2

0

0

0

-2

-2

-2

-4

-4

-4

(a)

R

= 0:1 rD , (b)

R

= 1:0 rD ,

( )

-1

R

z=rD 0

1

2

= 1:9 rD

FIG. 2: The first-order term [−e+ V (ρz, T )/T ] in units of q 2 /rD in Eqs. (5), (11), and (32) that contributes to the increments in n+ (ρz, R), σ+ (ρz, R), and u+ (ρz, R), as a function of the spatial coordinates ρ/rD and z/rD . Fig. 2(a) is for R = 0.1rD , Fig. 2(b) for R = 1.0rD , and Fig. 2(c) for R = 1.9rD .

We can write the above as N+ (R) = η(R) N+ (R → ∞)

(20)

where η(R) =

1 4π

Z

dζ



e−ζ− e−ζ+ − ζ+ ζ−

2

,

(21)

with ζ = r/rD , ζ± = |r ± R/2|/rD , η(0) = 0, and η(R → ∞) = 1. In Eq. (20), N+ (R → ∞) is the increment in the ¯ are far separated, number of positive medium particles when Q and Q  2 2 4π 3 q (22) r . N+ (R → ∞) = n0+ 3a2 rD T 3 D ¯ are close together, and the The increase in the total number of positive medium particles is small when Q and Q increase saturates when R reaches a few units of the Debye screening length rD . For our charge-neutral system, we obtained from Eq. (5) in a similar way N− (R) = N+ (R). Similarly, the entropy density of the medium particles is depleted near the static charge of the same sign, and is enhanced in the vicinity of the static charge of the opposite sign. When integrated over all spatial points, the depletion and the enhancement cancel to the first order. The second-order contributions always give a positive total entropy, Z Z  0 0 (23) T S± (R) = T dr σ± (r, R) − σ± = T σ± dr b2 [e± V (r, R)/T ]2 , which is small at small R and saturates at large R. We can write the above as T S± (R) = η(R) T S± (R → ∞)

(24)

where 0 T S± (R → ∞) = T σ± 3b2



q2 rD T

2

4π 3 r . 3 D

(25)

8

-2 0.8 0.6 0.4 0.2

=rD

z=rD

-2 -1 0.8 1 0.6 2 0.4 0.2

-1

0

z=rD 0

500

500

500

40

40

40

+V =T )2 (q 2 =r T )2 20 D 30

-2 0.8 1 0.6 0.4 2 0.2

30

30

20

20

10

10

10

0

0

0

(e

(a)

R

= 0:1 rD , (b)

R

= 1:0 rD ,

( )

R

-1

z=rD 0

1

2

= 1:9 rD

FIG. 3: The second-order term (e+ V (ρz, R)/T )2 in units of (q 2 /rD T )2 in Eqs. (5), (11), and (32) that contributes to n+ (ρz, R), σ+ (ρz, R), and u+ (ρz, R), as a function of the spatial coordinates ρ/rD and z/rD . Fig. 3(a) is for R = 0.1rD , Fig. 3(b) for R = 1.0rD , and Fig. 3(c) for R = 1.9rD .

A comparison of N± and T S± shows that N± (R) T S± (R) = = η(R). N± (R → ∞) T S± (R → ∞)

(26)

If we define N (R) = N+ + N− (R), and T S(R) = T S+ (R) + T S− (R), the increment of the total number and entropy ¯ (measured relative to the corresponding quantities in the of the medium particles due to the presence of Q and Q ¯ is absence of Q and Q), N (R) T S(R) = = η(R). N (R → ∞) T S(R → ∞)

(27)

Thus, the ratios T S(R)/T S(R → ∞), N± (R)/N± (R → ∞), and T S(R)/T S(R → ∞) [also U (R)/U (R → ∞) as we shall see in the next section] behave in the same way as a function of R. We show the behavior of η(R) = T S(R)/T S(R → ∞) for the Debye screening case in Fig 1(b) and it has the same shape as T S1 (R) obtained in the lattice gauge calculations shown in Fig. 1(a). One can therefore understand that as the result of constraining the system to be in contact with a thermal bath in the grand canonical ensemble, the medium particle numbers and entropy increase with increasing R to maintain a thermal equilibrium until they saturate at large separation. The thermal bath is therefore a participant in altering ¯ (See Section VI for a discussion the content of the medium particles, when the Q is separating from the antiquark Q. on the role of the thermal bath). V.

VARIATION OF THE INTERNAL ENERGY WITH R IN DEBYE SCREENING

¯ potential [27]. As Previously in quenched QCD, we find a relationship between the total internal energy and the Q-Q the problem of non-perturbative QCD so intrinsically complicated, a thorough understanding of such a relationship in an analogous, but not identical, problem in QED is worth having. We are therefore motivated to examine the mechanism of Debye screening due to the medium particles in QED. Of particular interest is to see whether the ¯ potential in Debye screening in QED resembles a similar relationship between the total internal energy and the Q-Q relationship in lattice gauge theory obtained previously in QCD [27].

9 ¯ separated by a distance R are screened by medium particles, the total internal energy Utotal (R) When Q and Q ¯ is the sum of the kinetic energy of the medium particles and the of the system of medium particles, Q, and Q ¯ when the medium particles have re-arranged themselves interaction energies of the the medium particles, Q, and Q, self-consistently,   Z Z g 2 p dp[f (r, p, R) + f (r, p, R)]p Utotal (R) = dr + − 2π 2   Z 1 R R + (28) dr e+ δ(r + ) + e− δ(r − ) + e+ n+ (r, R) + e− n− (r, R) V (r, R) 2 2 2 Z − dr(u0+ + u0− ), ¯ given by where u0± is the internal energy density of the medium in the absence of Q and Q u0± =

gT 4 7 ζ(4)Γ(4), 2π 2 8

(29)

¯ U0 (R) is measured relative to the total internal energy of the system in the absence of Q and Q, medium = Rand Utotal 0 0 dr(u+ + u− ). Using the solution of V (r, R) in Eq. (18) and excluding the infinite energies of a point source acting on itself, Eq. (28) gives q 2 e−µR q2 Utotal (R) = − − + R rD

Z

  dr {u+ (r, R) − u0+ } + {u− (r, R) − u0− } ,

(30)

where u± (r, R) are the internal energy density of the positive and negative charged medium particles in the presence ¯ given by of Q and Q   Z e± V (r, R) g 2 p + p dpf (r, p, R) u± (r, R) = ± 2π 2 2 e± V (r, R) ≡ hp + i. (31) 2 Using the local Fermi-Dirac distribution of Eq. (3), we obtain  u± (r, R) = u0± 1 − c1 [e± V ((r, R)/T ] + c2 [e± V (r, R)/T ]2 ,

(32)

where c1 and c2 are positive constants,

c1 =

Γ(3) 3 4 ζ(3)[Γ(4) − 2 ] 7 8 ζ(4)Γ(4)

= 0.7933,

(33)

c2 =

Γ(4) Γ(3) 1 2 ζ(2)[ 2 − 2 ] 7 8 ζ(4)

= 0.2895.

(34)

and

A comparison of u± (r, R) in Eq. (32) with n± (r, R) and u± (r, R) in Eqs. (5) and (11) indicates that the first-order and second-order contributions to u± (r, R) behave in the same way as those of n± (r, R) and σ± (r, R). The first-order increment of the internal energy density of the medium particles is suppressed near the static charge of the same sign, and is enhanced near the static charge of the opposite sign as shown in Fig. 2, while the second-order contributions to the internal energy are always positive as shown in Fig. 3. The first-order contributions cancel each other, when they are integrated over all the spatial points. However, the second-order contributions are always positive, and the integration of the second-order contributions always yield a positive quantity. The total medium internal energy of the positive and negative medium particles, measured relative to the corresponding quantities in the absence of Q ¯ are and Q, Z Z  U± (R) = dr u± (r, R) − u0± = n0± dr c2 [e± V (r, R)/T ]2 . (35)

10 ¯ the total internal energy in the presence of Q and Q, ¯ Relative to the total internal energy in the absence of Q and Q, is 0 Umedium(R) − Umedium = U+ (R) + U− (R).

(36)

Eq. (30) can therefore be written as Utotal (R) = −

q2 q 2 e−µR 0 + Umedium (R) − Umedium − R rD

(37)

From Eqs. (35), (19), and (21), the quantity U± (R) can be written as U± (R) = η(R), U± (R → ∞)

(38)

where U± (R → ∞) = u0± 3c2



q2 rD T

2

4π 3 r . 3 D

(39)

A comparison of N± and T S± shows that N± (R) T S± (R) U± (R) = = = η(R). U± (R → ∞) N± (R → ∞) T S± (R → ∞)

(40)

Consequently, if we define U (R) = U+ (R) + U− (R), we also have U (R) N (R) T S(R) = = = η(R). U (R → ∞) N (R → ∞) T S(R → ∞)

(41)

In this simple model of Debye screening, the ratios of T S(R)/T S(R → ∞), N (R)/N (R → ∞) and U (R)/U (R → ∞) are equal and their behavior is shown in Fig. 1(b). The entropy, total number, and internal energy of the medium ¯ is zero at R = 0 and increases as a particles (relative to the corresponding quantities in the absence of Q and Q) function of R until they saturate when the separation R reaches a few units of the Debye screening length. VI.

¯ SYSTEM IN DEBYE SCREENING ¨ SCHRODINGER EQUATION FOR Q-Q

As the dynamics of a quark and an antiquark in a quark-gluon plasma is very complicated, it is worth having a good ¯ in an understanding of the mechanism of screening and its effects on the interaction between a Q and an antiquark Q analogous, but not identical, problem. We are therefore motivated to examine the mechanism of Debye screening due ¯ potential in the to the medium particles in QED. Of particular interest is to obtain the relationship between the Q-Q Hamiltonian and the total internal energy Utotal , in order to find out whether this relationship resembles a similar relationship in lattice gauge theory obtained previously in Ref. [27]. ¯ system in a medium under Debye It is however somewhat tricky to determine the Hamiltonian for the Q and Q screening (and analogously, but not identically, under color-charge screening in QCD). We can follow the basic principles of statistical physics as described by Landau and Lifshitz [53]. Accordingly, we start with a closed system ¯ with medium particles and consider a small “subsystem” S that contains the Q, the Q, ¯ and the of a heavy Q and Q medium. As the number of medium particles of the whole closed system is very large, the number of medium particles contained in the small subsystem S can still be very large, and a statistical description of this small subsystem S is applicable. This subsystem S is not a closed system and it undergoes all kinds of interaction and medium particle exchanges with the complementary part S ′ of the whole system. We can describe this subsystem S to be in contact with a very large complementary part S ′ , which we can call a “thermal bath” in this connection. The QED system ¯ discussed in the last few sections or the QCD system in lattice gauge calculations, (the so-called “system” of Q, Q, and medium particles in contact with a thermal bath), corresponds in actual fact to the “subsystem” S out of the whole closed system S + S ′ . Under a perturbation of the subsystem S away from thermal equilibrium such as occurs in the displacement of Q ¯ the medium in the subsystem S will respond to the perturbation and will relax to a new state of thermal relative to Q, equilibrium after a certain relaxation time, trelax (S). For example, from the results concerning the medium entropy ¯ obtained in the last few and number contents in the subsystem S as a function of the separation between Q and Q

11 ¯ the medium particles sections (Fig. 1), we know that under a displacement of the relative separation of the Q and Q, ′ will exchange between S and the thermal bath S in order to make the subsystem S under thermal equilibrium. In this case, the relaxation time trelax (S) corresponds to the exchange of medium particles through imaginary boundaries between S and the thermal bath S ′ . Relaxation time grows smaller as the subsystem S decreases in size [53]. For a ¯ in the small subsystem S, this relaxation time trelax (S) can be very short. On the other hand, for heavy Q and Q ¯ relative motion, tQQ¯ , can be relatively long because of the large mass of Q and Q. ¯ The subsystem, the period of Q-Q period tQQ¯ can be so much greater than the medium relaxation time trelax (S), tQQ¯ >> trelax (S), that the medium can be approximately considered as reaching a state of thermal equilibrium approximately instantaneously, at any time ¯ For the medium particles, this is just during the (supposedly slow) motion of the heavy quark Q and antiquark Q. the Born-Oppenheimer approximation for the description of the states of the medium particles in the subsystem S, as presented in the last few sections and used in lattice gauge calculations to obtain the medium particle configurations in QCD. ¯ in the In what sense can energy and entropy be considered conserved under a periodic motion of the Q and Q medium? The whole closed system consists of the subsystem S ′ and S, and the subsystem S is not a closed system. ¯ in the subsystem S, then From the results of the last few sections, we know that if we move the Q closer relative to the Q the medium particles (and its entropy and energy contents) will move from the subsystem S into the complementary ¯ part S ′ so as to maintain thermal equilibrium in the subsystem S. When we move the Q farther relative to the Q, ′ then the medium particles, entropy, and energy contents will move from the complementary system S back into the ¯ the subsystem S so as to maintain thermal equilibrium in the subsystem S. For slow periodic motion of Q and Q, motion can be so slow that the exchange of medium particles between S and S ′ can be approximated as taking place ¯ can thus be adiabatic in with no excitation of the medium. In this sense, the idealized periodic motion of Q and Q the lowest order with an “adiabatic” exchanging the energy content and the entropy content of the medium particles back and forth between the subsystem S and the complementary system S ′ . Additional interactions of the bound ¯ states with the medium particles that lead to the non-adiabatic excitation of both objects can then be periodic Q-Q considered in higher-order approximations. ¯ experience the interactions from all medium Thus, in this adiabatic picture of tQQ¯ >> trelax (S), the Q and Q particles which adjusts themselves (within a relaxation time which is taken to be so small as to be approximately ¯ The potential energy of the Q and Q ¯ at a instantaneous) at all instances of the dynamical motion of the Q and Q. ¯ is half of the integral of the product of the local point charges of the separation r in the Hamiltonian for the Q and Q ¯ with their local potentials V (r, R) arising from a self-consistent rearrangement of all particles when Q and Q and Q ¯ Q are at the separation R, excluding the infinite self-energy contributions. In addition to the potential energy of the ¯ the Hamiltonian for the Q-Q ¯ system consists also of the kinetic energy of Q and Q. ¯ The Hamiltonian for Q and Q, ¯ the Q-Q system is therefore given by   Z p2Q¯ p2Q 1 R R H = + + dr e+ δ(r + ) + e− δ(r − ) V (r, R) 2mQ 2mQ¯ 2 2 2 p2Q¯ p2Q 1 = + + [e+ V (r, R)|r=−R/2 + e− V (r, R)|r=R/2 ]. 2mQ 2mQ¯ 2

(42)

Upon making the change of the variables from pQ and pQ¯ to the center-of-mass momentum PCM = pQ + pQ¯ and the relative momentum pR = (pQ − pQ¯ )/2, and using the solution V (r, R) of Eq. (18), we find from the above equation H=

q 2 e−µR p2 q2 P2CM + R − − , 2(mQ + mQ¯ ) 2µred R rD

(43)

where µred = mQ mQ¯ /(mQ + mQ¯ ) is the reduced mass. The Hamiltonian then separates into H = HCM + HR where HCM is the Hamiltonian for the free motion of the composite two-body system and HR is the Hamiltonian for the ¯ relative motion of Q and Q, HR =

p2R q 2 e−µR p2R q2 − ≡ + UQQ¯ (R) − 2µred R rD 2µred

(44)

¯ system under screening by the medium, UQQ¯ (R), From the above equation, we recognize that the potential for the Q-Q is given by UQQ¯ (R) = −

q2 q 2 e−µR − . R rD

¯ potential UQQ¯ (R) is just the Debye screening potential plus an R-independent constant term. The QQ

(45)

12 ¯ the Hamiltonian formulated in Eq. (42) indeed gives Based on the “adiabatic” picture of the motion of Q and Q, correctly the Hamiltonian with the Debye screening potential. The R-independent term −q 2 /rD is also an important ¯ Hamiltonian for the case of small µ representing part of the screening contribution. We note that if we expand the Q-Q the screening effects, then Eq. (44) becomes, HR ∼

p2R q 2 (1 − µR) q 2 q2 p2R − − , − = 2µred R rD 2µred R

(46)

which is the same Hamiltonian as that of the unscreened case. Thus, we reach the interesting result that in the lowest order of the screening parameter µ = 1/rD , a properly calibrated Hamiltonian of a system under screening, with the shift of the level of the potential, −q 2 /rD , is the same Hamiltonian without screening. In practical terms, if we calculate ¯ system without screening, we expect that within the lowest order of the screening parameter µ, the mass of a Q-Q the mass eigenvalue of the system to be nearly unchanged when screening is present. Numerical calculations of the mass of bound L = 0 charmonium using the potential model of Ref. [27] indeed shows that the absolute value of the charmonium L = 0 mass changes only very slightly as a function of temperature up to 1.5Tc [30]. ¯ From the above discussions in the simple case of Debye screening, we observe that the potential between Q and Q, UQQ¯ (R), differs from the total internal energy Utotal (R). Because of Eqs. (45) and (37), they are related by 0 UQQ¯ (R) = Utotal (R) − [Umedium (R) − Umedium ].

(47)

0 It is therefore necessary to subtract out the change of the medium internal energies [Umedium(R) − Umedium ] from the ¯ total internal energy Utotal (R) to obtain the Q-Q potential UQQ¯ (R) in the grand canonical ensemble. This conclusion for Debye screening supports a similar conclusions in the analogous lattice gauge theory, where we have proved in Eq. (11) of Ref. [27], (1)

UQQ¯ (R, T ) = U1 (R, T ) − [Ug(1) (R, T ) − Ug0 (T )].

(48)

¯ and [Ug(1) (R, T ) − Ug0 (T )] is In the above equation, the superscript (1) refers to the color-singlet state of Q and Q, ¯ the increment of gluon energy due to the presence of Q and Q. From the above analysis, we conclude that the relationship of Eq. (48) in Ref. [27] is a rather general result for heavy particles under screening in the grand canonical ensemble. We can understand the results of Eq. (48) [or similarly (47)] from another viewpoint. In a standard description of a ¯ in a medium, we simplify the dynamics by considering first the (QQ) ¯ states and the deconfined medium states (QQ) separately as independent unperturbed states. We then include their mutual excitations as perturbative couplings. ¯ states should be obtained without Thus, in the lowest-order description without perturbative couplings, the (QQ) the excitation of the medium states of deconfined real gluons and vice versa. In the quenched approximation, the deconfined real gluons in the quark-gluon plasma in the Feynman diagram language are those represented by lines (1) with external legs. The change of the medium internal energies, [Ug (R, T ) − Ug0 (T )] in Eq. (48), represents the excitation of the internal energy states of the deconfined real gluon medium when the separation between the Q ¯ changes in the grand canonical ensemble. As the unperturbed states of the Q-Q ¯ relative motion should and the Q be calculated without the excitation of the medium states of deconfined real gluons, we therefore need to subtract ¯ potential the change of the real gluon internal energy from the total internal energy in Eq. (48) to obtain the Q-Q (1) ¯ changes. UQQ¯ (R, T ), when the separation between the Q and the Q In the quenched approximation, the subtraction of this change in the internal energy of real gluons as a function of R does not mean that both the real and the virtual gluon degrees of freedom are frozen. Only the real gluon ¯ bound states of relative excitation energy degrees of freedom are frozen when we calculate the unperturbed Q-Q motion, for reasons we have just given. On the other hand, virtual gluons, which in the Feynman diagram language are represented by gluon lines with the two end points of each line joining onto other quarks and gluons, change their ¯ changes. These virtual gluons mediate the interaction configurations as the separation R between the Q and the Q ¯ between the Q and the Q. The changes in the virtual gluon configurations modify the interaction between the Q ¯ resulting in the screening of the Q-Q ¯ potential. This type of virtual gluon excitation is not frozen and is and the Q, ¯ ¯ relative motion. included in the calculation of the Q-Q potential and the evaluation of bound states of the Q-Q The reconfiguration of these virtual gluons can take place in either an adiabatic manner or in the opposite “diabatic” manner depending on whether the time scale of the relaxation of these virtual gluons is short compared to the time ¯ relative motion. As we explained earlier, the adiabatic description is appropriate for scale of the period of the Q-Q ¯ is slow. This is indeed confirmed by the successful very heavy quark pairs when the relative motion of the Q and the Q ¯ pair as the heavy quark Q-Q ¯ potential, U (1)¯ (R) at T = 0 identification of the lattice free energy for a static Q-Q QQ [Bali et al. Phys. Rev. D56 2566 (1997)]. For this case of T = 0, there are no free gluons nor energy excitations of

13 ¯ changes, and the total free energy of the free gluon medium states when the separation R between the Q and the Q system is equal to the total internal energy of the system. The real gluon energy degrees of freedom are absent and frozen but the virtual gluon degrees of freedom adjust themselves as R changes. The question whether an adiabatic or a diabatic picture is a more appropriate description for the potential arises also in the NN and meson-meson problems at T = 0 in lattice gauge theory. However aside from the question of ¯ NN, and meson-meson potentials differ in their different degrees of freedom and the methods of adiabaticity, the Q-Q, ¯ potential studied here is a (two-body)-plus-(deconfined medium) problem, while the NN and calculations. The Q-Q meson-meson potentials at T = 0 are (six-body)-plus-(virtual gluons) and (four-body)-plus-(virtual gluons) problems respectively. At T = 0, a wave function treatment of the lattice gauge correlator results in the correct repulsive potential for the NN potential at short distances [55], while an “adiabatic” potential treatment without using the lattice wave function gives flat NN and meson-meson potentials [56]. On the contrary, however, another “adiabatic” lattice gauge meson-meson potential calculation at T = 0 gives repulsive and attractive inner cores when different internal degrees of freedom of the light quarks are taken into account [57]. Furthermore, the lattice gauge wave function method of [55] may need additional justifications as questions have been raised in Appendix A of [57] concerning its lattice wave function assumption. While the work of [55] appears to give a correct description, much work remains to be carried out to sort out the differences of the lattice gauge calculations of [55], [56], and [57]. It remains another separate additional question how one can obtain definitive conclusions on the adiabaticity or diabaticity of the (twobody)-plus-(deconfined medium) potential at T > Tc from these (six-body)- and (four-body)-plus-(virtual gluon) problems at T = 0. As many unanswered questions remains to be resolved, the results of [55] cannot yet be used, for ¯ potential examined here. the present time at least, to draw conclusions on the adiabaticity or diabaticity of the Q-Q Nevertheless, the exploration of the relationship between adiabaticity and the shape of the relative wave function is an interesting subject for future investigations. VII.

¯ POTENTIAL FROM U1 AN APPROXIMATE METHOD TO SEPARATE OUT THE Q-Q

From the simple model of Debye screening, we observe that up to the first order of eV /T , the internal energy of the medium does not change, but up to the second order the internal energy increases with an increasing separation ¯ This increase arises from the fact that the thermal equilibrium attained through the contact with between Q and Q. a thermal bath in a grand canonical ensemble constrains the occupation numbers of the medium particles, and this newly re-arranged distribution leads to an increase in the number, the entropy, and the internal energy of the medium, as a function of increasing R. Returning now to QCD lattice gauge calculations and noting its similarities with Debye screening of Coulomb charges, we should therefore expect that the number, the entropy, and the internal energy of the gluon medium ¯ Indeed, as shown for lattice gauge calculations should likewise increase as function of increasing R between Q and Q. at a fixed temperature in Fig. 1(a), there is an increase in the entropy of the system as R increases, similar to the analogous Debye screening case shown in Fig. 1(b). ¯ potential from Having understood the behavior of various thermodynamic quantities, we wish to extract the Q-Q (1) lattice gauge results. The most reliable way is to carry out additional lattice gauge calculations to obtain Ug (R) ¯ potential is then the difference of U1 (R) and Ug(1) (R) − Ug0 , as given by Eq. (48). As Ug(1) (R) and and Ug0 . The Q-Q Ug0 in lattice gauge calculations are not yet available, we will try to use another piece of lattice gauge data to obtain ¯ potential, as least approximately. the Q-Q 0 We note that in the Debye screening case Umedium (R) − Umedium is proportional to T S(R), and in the lattice gauge calculations the quantity T S1 (R, T ) has been calculated. We can look for a similar relationship between the gluon (1) internal energy and the gluon entropy for the quark-gluon plasma. If we succeed in relating Ug (R, T ) − Ug0 to (1) ¯ potential, U ¯ , can be determined from U1 (R, T ) by subtraction using Eq. (48). T S1 (R, T ), then the Q-Q QQ The subtraction can be carried out by noting that locally the quark-gluon plasma internal energy density ǫ is related to its pressure p and entropy density σ by the First Law of Thermodynamics, ǫ = T σ − p,

(49)

and the quark-gluon plasma pressure p is also related to the plasma energy density ǫ by the equation of state p(ǫ) that is presumed known by another lattice gauge calculation. Thus, by expressing p as (3p/ǫ)(ǫ/3) with the ratio a(T ) = 3p/ǫ given by the known equation of state, the plasma internal energy density ǫ is related to the entropy density T σ by ǫ=

3 T σ. 3 + a(T )

(50)

14 This is just (1)

dUg dV

=

d 3 3 + a(T ) dV

Z

dr T (σ − σ0 + σ0 ),

(51)

¯ Noting that the entropy of the medium for the color-singlet where σ0 is the entropy R density in the absence of Q and Q. R ¯ pair is T S1 = drT (σ − σ0 ) and Ug0 is related to dr T σ0 , the above equation leads to Q-Q (1)

T dS1 (R, T ) 3 d[Ug (R, T ) − Ug0 (T )] = , dV 3 + a(T ) dV

(52)

and the plasma internal energy integrated over the volume is given by Ug(1) (R, T ) − Ug0 (T ) =

3 T S1 (R, T ). 3 + a(T )

(53)

But T S1 (R, T ) has already been obtained as U1 (R, T ) − F1 (R, T ). The plasma internal energy is therefore equal to Ug(1) (R, T ) − Ug0 =

3 [U1 (R, T ) − F1 (R, T )]. 3 + a(T )

(54)

(1)

¯ potential, U ¯ , as determined by subtracting the above plasma internal energy from U1 , is then a linear The Q-Q QQ combination of F1 and U1 given by [27], (1)

W1 (R, T ) ≡ UQQ¯ (R, T ) =

3 a(T ) F1 (R, T ) + U1 (R, T ), 3 + a(T ) 3 + a(T )

(55)

(1)

where for brevity of notation we have renamed UQQ¯ (R, T ) as W1 (R, T ) and we can define the coefficient of F1 , fF = 3/(3 + a(T )), as the F1 fraction, and the coefficient of U1 , fU = a(T )/(3 + a(T )), as the U1 fraction. The (1) potential UQQ¯ is approximately F1 near Tc and is approximately 3F1 /4 + U1 /4 for T > 1.5Tc [27]. VIII.

¯ POTENTIALS COMPARISON OF DIFFERENT Q-Q

In the spectral function analyses, the widths of many color-singlet heavy quarkonium states broaden suddenly at various temperatures [21, 22, 35]. In the most precise calculations for J/ψ using up to 128 time-like lattice slices, the spectrum has a sharp peak for 0.78Tc ≤ T ≤ 1.62Tc and a broad structure with no sharp peak for 1.70Tc ≤ T ≤ 2.33Tc [21]. The spectral peak at the bound state has the same structure and shape at 0.78Tc as it is at 1.62Tc. If one can infer that J/ψ is stable and bound at 0.78Tc, then it would be reasonable to infer that J/ψ is also bound and stable at 1.62 Tc . The spectral function at 1.70Tc has the same structure and shape as the spectral function at 2.33Tc. If one can infer that J/ψ is unbound at 2.33Tc, then it would be reasonable to infer that J/ψ become already unbound at 1.70 Tc . We can define the spontaneous dissociation temperature of a quarkonium as the temperature at which the quarkonium changes from bound to unbound and dissociates spontaneously. Thus, from the shape of the spectral functions, the temperature at which the width of a J/ψ quarkonium broadens suddenly from 1.62Tc to 1.70Tc corresponds to the J/ψ spontaneous dissociation temperature. Spontaneous dissociation temperatures for χc and χb have been obtained in [22, 35]. We list the heavy quarkonium spontaneous dissociation temperatures obtained from spectral analyses in (1) quenched QCD in Table I. They can be used to test the potential models of W1 (≡ UQQ¯ ), F1 , and U1 . Table I. Spontaneous dissociation temperatures obtained from different analyses. Quenched QCD States Spectral Analyses W1 F1 J/ψ, ηc 1.62-1.70Tc† 1.62 Tc 1.40 Tc χc below 1.1Tc♮ unbound unbound ′ ′ ψ , ηc unbound unbound Υ, ηb 4.1 Tc 3.5 Tc χb 1.15-1.54Tc♯ 1.18 Tc 1.10 Tc Υ′ , ηb′ 1.38 Tc 1.19 Tc † Ref.[21], ♮ Ref.[22], ♯ Ref.[35]

U1 2.60 Tc 1.18 Tc 1.23 Tc ∼ 5.0 Tc 1.73 Tc 2.28 Tc

Full QCD (2 flavors) W1 F1 U1 1.42 Tc 1.21 Tc 2.22 Tc 1.05 Tc unbound 1.17 Tc unbound unbound 1.11 Tc 3.40 Tc 2.90 Tc 4.18 Tc 1.22 Tc 1.07 Tc 1.61 Tc 1.18 Tc 1.06 Tc 1.47 Tc

¯ potentials in quenched QCD, we use the free energy F1 and the internal energy U1 obtained by To evaluate the Q-Q

15 Kaczmarek et al. [33] where F1 and U1 can be parametrized in terms of a screened Coulomb potential with parameters shown in Figs. 2 and 3 of Ref. [27]. For the ratio a(T ) from the plasma equation of state in Eq. (55), we use the quenched equation of state of Boyd et al. [58] for quenched QCD. The quantity a(T ) = 3p/ǫ and the U1 and F1 fractions as a function of T are shown Fig. 1(b) and Fig. 1(c) of Ref. [27] respectively. These quantities allows the (1) specification of the W1 ≡ UQQ¯ potential as a function of temperature. Using quark masses mc = 1.41 GeV and mb = 4.3 GeV, we can calculate the binding energies of heavy quarkonia and their spontaneous dissociation temperatures using different potentials in quenched QCD. As a function of temperature, the binding energies and wave functions of charmonia have been presented in Fig. 6 and 7 of Ref. [27] respectively. The bounding energies of bottomia have been presented p in Figs. 8 and 9, and the wave functions in Fig. 10 of Ref. ¯ separation hR2 i of L = 0 charmonium calculated with mc = 1.41 GeV in [27]. We show the root-mean-square Q-Q the W1 (R) potential in quenched QCD as the solid curve in Fig. 4. As one observes, the root-mean-squared separation RRMS is about 1 fm for T ∼ Tc and it increases to about 4 fm at T = 1.6Tc before it becomes unbound. The large separation is expected for systems with a weak binding, in analogous to the halo nuclei with neutrons in weak binding observed in nuclear physics [59]. There is the question whether charmonium with such a large separation between Q ¯ may survive in QGP. The dissociation cross sections for these quarkonia by collision with gluons have been and Q calculated and found to be a function of the gluon collision energy, as shown in Fig. 13 of Ref. [27]. 5

Full QCD (2 flavors)

(fm)

2

Quenched QCD

2

3

√< R >

4

1

0

1

1.2

1.4 T/TC

1.6

1.8

¯ separation of L = 0 charmonium as a function of T /Tc in quenched QCD (solid curve), FIG. 4: The root-mean-squared Q-Q and in full QCD with two favors (dashed curve).

From the binding energy of a quarkonium as a function of temperature, one can obtain the temperature at which the quarkonium binding energy vanishes. This is the temperature for the spontaneous dissociation of the quarkonium, as the quarkonium at this temperature will dissociate spontaneously. We list in Table I the heavy quarkonium spontaneous dissociation temperatures calculated with the W1 potential, the F1 potential, and the U1 potential, in quenched QCD. The J/ψ and χb spontaneous dissociation temperatures obtained with the W1 potential in quenched QCD are found to be 1.62Tc and 1.18Tc respectively. Spectral analyses in quenched QCD give the spontaneous dissociation temperature of 1.62-1.70Tc for J/ψ [21] and 1.15-1.54Tc for χb [35]. Thus, spontaneous dissociation temperatures obtained with the W1 potential agree with those from spectral function analyses. This indicates that the W1 potential, ¯ potential for studying the defined as the linear combination of U1 and F1 in Eq. (55), may be the appropriate Q-Q stability of heavy quarkonia in quark-gluon plasma. IX.

¯ POTENTIAL FOR FULL QCD WITH TWO FLAVORS Q-Q

The interaction energy between a heavy quark and a heavy antiquark in the color-singlet state in two-flavor full QCD was studied by Kaczmarek and Zantow [34]. In full QCD with 2 flavors, F1 and U1 can be represented by a color-Coulomb interaction at short distances and a completely screened, constant, potential at large distances as given

16 in Ref. [28], although other alternative representations have also been presented [36, 37]. The transitional behavior linking the two different spatial regions can be described by a radius parameter r0 (T ) and a diffuseness parameter d(T ), as in the Wood-Saxon shape potential in nuclear physics, {F1 , U1 }(R, T ) = −

4 αs (T ) f (R, T ) + C(T )[1 − f (R, T )], 3 R

(56)

1 . exp{(R − r0 (T ))/d(T )} + 1

(57)

where f (R, T ) =

r0 (fm)

C(T) (GeV)

In principle, it is necessary to specify only the temperature dependence of F1 (R, T ) as the internal energy U1 (R, T ) can be obtained from F1 and its derivative with respect to T . In practice, as Kaczmarek and Zantow [34] have obtained U1 (R, T ) by a careful numerical differentiation, it is convenient to parametrize the internal energy in the above simple form for practical calculations.

1.0

Parameters for F1 (R,T)

0.5 0.0 0.6 0.4 0.2

d (fm)

0.0 0.3 0.2 0.1 0.0

1.0

1.5

2.0

2.5 T/Tc

3.0

3.5

4.0

FIG. 5: The parameters C, r0 , and d for the color-singlet free energy F1 (R, T ) in two-flavor QCD as given in Eq. (56).

In searching for the coupling constant αs that fits the lattice quantities, we found that the value of αs centers around 0.3. The fit to the lattice gauge quantities does not change significantly whether we allow αs to vary. It is convenient to keep the value of αs to be 0.3 so that there are only three parameters for each temperature. For the free energy F1 (R, T ) in two-flavor QCD [34], the set of parameters C, r0 , and d are shown in Fig. 5, and the corresponding fits to F1 are shown in Fig. 6. For the internal energy U1 (R, T ) in two-flavor QCD [34], the set of parameters C, r0 , and d are shown in Fig. 7, and the corresponding fits to the lattice gauge internal energy U1 results shown in Fig. 8. If the thermodynamical quantity F1 or U1 are treated as a potential, then the quantity C(T ) is an approximate measure of the depth of the potential measured from the flat potential surface at large distances relative to the potential well at short distances. For the free energy F1 , the C(T ) parameter has the value of about 1 GeV at T ∼ 0.8Tc , and it decreases to 0.5 GeV at Tc . The free energy as a potential will has a well depth of about 0.5 GeV for T close to Tc and the well depth decreases to about 0.1 GeV at T ∼ 2Tc . One notes that there is a significant change in the slopes of C(T ) at T ∼ Tc for F1 . As a consequence, the parameter C(T ) for U1 exhibits a peaks at T ∼ Tc . The transitional radius r0 for F1 decreases gradually from about 0.6 fm to about 0.15 fm and the diffuseness parameter d decreases slowly from 0.3 fm to about 0.15 fm, as temperatures decreases from 0.7Tc to 4Tc .

17

T/Tc 0.76 0.81 0.87 0.90 0.96 1.00 1.02 1.07 1.11 1.16 1.23 1.36 1.50 1.65 1.81 1.98 4.01

1.2 1.0

F1(R,T) (GeV)

0.8 0.6 0.4 0.2 0.0

-0.2 -0.4 0.0

0.5

1.0

1.5 R (fm)

2.0

2.5

C(T) (GeV)

FIG. 6: The symbols represent the color-singlet free energy, F1 (R, T ), for two-flavor QCD [34], and the curves are the fits using the screened potential, Eq. (56), with parameters given in Fig. 5.

4.0

Parameters for U1 (R,T)

3.0 2.0 1.0

d (fm)

r0 (fm)

0.0 1.0 0.5 0.0 0.4 0.2 0.0

1.0

1.5

2.0 T/Tc

2.5

3.0

FIG. 7: The parameters C, r0 , and d for the color-singlet internal energy U1 (R, T ) as given in Eq. (56).

For the internal energy U1 , the parameter C(T ) is quite large, attaining the value of about 3 GeV for T close to Tc . This indicates that if U1 is used as a potential, the potential depth at temperatures close to the transition temperature is of order 3 GeV, which is a very deep potential indeed. The parameter C(T ) decreases to about 0.8 GeV when the temperature exceeds about 1.5 Tc . The transition radius r0 is about 1 fm for T close to 0.8 Tc , and it decreases to about 0.2 fm for T ∼ 4Tc . The diffuseness parameter d(T ) for the internal energy decreases substantially at temperatures below Tc , but maintains a relatively constant values of 0.1 to 0.2 fm for T greater than Tc . The comparison in Fig. 6 and 8 shows that the free energy F1 and the internal energy U1 with the set of parameters in Figs. 4 and 6, adequately describe the lattice-gauge data and can be used to calculate the eigenvalues and eigenfunctions of heavy quarkonia. ¯ potential is a linear combination of U1 and F1 with coefficients depending on the In our description, the Q-Q equation of state. The equation of state in full QCD with two flavors has been obtained by Karsch et al. [47]. We show their results of ǫ/T 4 and 3p/T 4 in Fig. 9(a). The ratio a(T ) = 3p/ǫ and the U1 and F1 fractions as a function of T are shown in Figs. 8(b) and 8(c) respectively. Similar to the case of quenched QCD, the F1 fraction is close to

18 T/Tc 0.98 0.93 0.88 0.78

U1(R,T) (GeV)

4.0 3.0 2.0 1.0 0.0

T/Tc 1.01 1.04 1.09 1.11 1.13 1.19 1.29 1.43 1.57 1.89 2.99

U1(R,T) (GeV)

4.0 3.0 2.0 1.0 0.0 0.0

0.5

1.5 1.0 R (fm)

2.0

FIG. 8: The symbols represent U1 (R, T ) for two-flavor QCD obtained by Kaczmarek et al. [34] and the curves are the fits using Eq. (56), with parameters given in Fig. 7.

unity near Tc and it decreases to 3/4 at large temperatures while U1 fraction is nearly zero at Tc and it increases to ¯ potential for bound about 1/4 at high temperatures. These quantities, together with F1 and U1 specifies the Q-Q state calculations. X.

HEAVY QUARKONIA IN QUARK-GLUON PLASMA

In full QCD with two flavors, the transition temperature is Tc = 202 MeV [34]. To calculate the charmonium energy levels, we employ a quark mass mc = 1.41 and mb = 4.3 GeV. Energy levels of charmonium states calculated with different potentials in full QCD with two flavors are shown in Fig. 10 as a function of the temperature in units of Tc . The J/ψ and ηc states are weakly bound and they dissociate at 1.21 Tc in the F1 potential, at 1.42Tc in the W1 potential, and at 2.22Tc in the U1 potential. The χc state dissociates below Tc in the F1 potential, at 1.05Tc in the W1 potential, and at 1.17Tc in the U1 potential. At temperatures slightly greater than Tc , they are weakly bound in the F1 potential but are strongly bound in the U1 potential, with a binding energy of about 0.7 GeV at 1.1Tc . The binding of the states in the W1 potential lies between these two limits. The dissociation temperature for J/ψ and ηc in full QCD with two flavors have been examined in the spectral function analysis by Aarts et al. [60]. As the calculations have been carried out only with a small lattice volume and a small set of statistics, there are possible systematic uncertainties which prevented a precise determination of the pseudocritical temperature. Preliminary results indicate that the J/ψ state may be bound up to about 2.Tc [60]. More definitive results will await a greater lattice volume and larger statistics. We also carry out the analysis of bottomium 1s, 2p, and 2s states. Fig. 11 gives the state energies as a function of T /Tc for different potentials. The eigenenergies of Υ and ηb in two-flavor QCD are about -0.1 GeV at T = 1.1 Tc in the F1 potential, and are about -1.0 GeV in the U1 potential. The eigenenergies in the W1 lie in between those of the F1 and U1 potentials. These states dissociate spontaneously at 2.9Tc in the F1 potential, 3.40 in the W1 potential, and about 4 to 5Tc in the U1 potential. In full QCD with two flavors, the χb state dissociates at 1.07Tc in the F1 potential, at 1.22 in the W1 potential, and at 1.61 in the U1 potential. The Υ′ and ηb′ state dissociates at 1.06Tc in the F1 potential, at 1.18 in the W1 potential, and at 1.47 in the U1 potential.

10.0 8.0 6.0 4.0 2.0 0.0 1.0 0.8 0.6 a(T) ≡ 3p/ε 0.4 0.2 0.0 1.0 0.8 fF 0.6 0.4 fU 0.2 0.0 1.5 1.0 2.0

4

ε/T 4 3p / T

a(T)

(a)

(b)

fF or fU

4

ε/T or 3p/T

4

19

(c)

2.5 T / Tc

3.0

3.5

FIG. 9: The equation of state in full QCD with two flavors: (a) the quantity ǫ/T 4 and 3p/T 4 , (b) the ratio a(T ) = 3p/ǫ, and ¯ potential U (1)¯ . (c) the U1 and F1 fractions in the Q-Q QQ

ε (GeV)

0.0

F1 W1 U1

-0.2

-0.4

J/ψ , ηc states mc=1.41GeV

χc

ψ’

(b)

(c)

-0.6

(a) -0.8 1.0

1.5 T/Tc

2.0

1 1.2 1.4 1 1.2 1.4 T/Tc T/Tc

FIG. 10: Energy levels of charmonium in the quark-gluon plasma as a function of temperature calculated with the F1 (R, T ), W1 (R, T ), and U1 (R, T ) potentials in two-flavor QCD. Fig. 10(a) is for the J/ψ and ηc state, Fig. 10(b) is for the χc state, and Fig. 10(c) is for the ψ ′ state.

In Table I we list the dissociation temperatures of different quarkonia obtained in full QCD with two flavors. A comparison of the dissociation temperatures from the quenched QCD and full QCD with two flavors show the effects of the dynamical quarks. Dynamical quarks increases the degree of screening, but at the same time, lowers the phase transition temperature. They lead to a more diffused potential with a greater screening length. As a consequence, the binding energy of the 1s states is lowered and the dissociation temperature decreases. We can chooses the W1 ¯ pair, as it gives the best agreement with spectral function potential as the more appropriate potential for the Q-Q analysis in quenched QCD. For this W1 potential, the dissociation temperature decreases from 1.62Tc in quenched

20

0.0

ε (GeV)

-0.2 -0.4

F1 W1 U1

-0.6 -0.8

Υ , ηb states

-1.0

χb

Υ’

(b)

(c)

mb= 4.3 GeV

-1.2 -1.4

(a)

-1.6 -1.8

1

2

3 T/Tc

4 1

2 1 T/Tc

2 T/Tc

FIG. 11: Energy levels of bottomium in the quark-gluon plasma as a function of the temperature calculated with the F1 (R, T ), W1 (R, T ), and U1 (R, T ) potentials in full QCD with two flavors. Fig. 11(a) is for the Υ and ηb state, Fig. 11(b) is for the χb state, and Fig. 11(c) is for the Υ′ state.

QCD to 1.42Tc in full QCD with two flavors. The effects of the dynamical quark in full QCD leads to a slightly weaker binding for J/ψ in the plasma. For the χ states, the effects of the the additional quark screening does not modify the dissociation temperature substantially. The additional screening tends to move the centrifugal barrier for the l = 1 state to a smaller radial distance with a slightly higher barrier, resulting in a very slight increase in the dissociation temperature. XI.

QUARK-DRIP LINES IN QUARK-GLUON PLASMA

¯ pair, we consider the quark mass mQ as a variable and evaluate To examine the stability of a color-singlet Q-Q the spontaneous dissociation temperature as a function of the reduced mass µred = mQ mQ¯ /(mQ + mQ¯ ). The quark drip lines calculated with the F1 , W1 , and U1 potentials in quenched QCD are shown in Fig. 12. A state is bound in the (T /Tc , µred ) space above a drip line and is unbound below the drip line. Spectral function analysis gives the spontaneous dissociation temperature of 1.62-1.70Tc for J/ψ [21, 22] and 1.15-1.54Tc for χb [22, 35]. If one takes the charm quark mass to be 1.41 GeV and the bottom quark mass to be 4.3 GeV, the spectral function results can be represented by the solid-circle symbols in Fig. 12. They fall on the drip line curves obtained with the W1 potential, ¯ potential to use for bound state problems. indicating that the W1 is the appropriate Q-Q In quenched QCD, however, the quark-gluon plasma is assumed to consist of gluons only. As dynamical quarks may provide additional screening, it is necessary to consider the case with dynamical quarks. Accordingly, we use the F1 , W1 , and U1 potentials evaluated in full QCD with 2 flavors [34, 47] to determine the drip lines in Fig. 13. The drip lines for the U1 potential lies lower than that of the W1 , which in turn lies lower than the drip line of the F1 potential. In comparison with quenched QCD results, the 1s drip line in full QCD is shifted to lower temperatures while the 1p drip line in full QCD is only slightly modified. We shall use the results from the W1 potential in full QCD with two flavors to discuss the question of quarkonium stability in quark-gluon plasma, as the W1 potential has been found to give results in agreement with spectral function analyses in quenched QCD. For heavy quarkonia, results in Table I and Fig. 13 obtained with the W1 potential in full QCD with two flavors indicate that J/ψ, χc , Υ, χb , and Υ′ may be bound in the plasma up to 1.42Tc, 1.05Tc, 3.40Tc, 1.22Tc, and 1.18Tc respectively. The variation of the drip lines with the reduced mass allows us to examine the stability of quarkonia containing quarks of various masses. We need to know the effective masses of different quarks in the quark-gluon plasma. Due to its strong interaction with other constituents, a light quark becomes a dressed quasiparticle and acquires a large quasiparticle mass. In the low temperature region where the spontaneous chiral symmetry breaking occurs with

21

1p drip line

µred (GeV)

2.0

Quenched QCD F1 W1

U1

1.5

1s drip line F1

1.0

W1 W1 F1 U1

U1

0.5 0.0 1.0

1.5

2.0

2.5 3.0 T/Tc

3.5

4.0

FIG. 12: The quark drip lines in quenched QCD calculated with the F1 , W1 , and U1 potentials. The symbols represent results from lattice gauge spectral function analyses.

F1

µred (GeV)

2

1p drip line W1

1s drip line

U1

F1

1.5

W1

1

W1 F1 U1

U1

0.5

Two-Flavor QCD

0

1

1.5

2

2.5 T/Tc

3

3.5

4

FIG. 13: The quark drip lines in 2-flavor QCD calculated with the W1 , F1 and U1 potentials.

¯ = ¯ hψψi 6 0, the quasiparticle mass is mq ∼ [|ghψψi|+(current quark mass)], where g is the strong coupling constant ¯ and hψψi the quark condensate [43, 61, 62]. This quasiparticle mass is the origin of the constituent-quark mass in non-relativistic constituent quark √ models [43, 62, 63, 64]. In the high temperature perturbative QCD region, the quasiparticle mass is mq ∼ gT / 6, which is of the order of a few hundred MeV [65]. ¯ decreases gradually as the temperature As the restoration of chiral symmetry is a second order transition, hψψi ¯ will likewise decrease gradually from increases beyond Tc . The light quark quasiparticle mass associated with hψψi the constituent-quark mass value to the current-quark mass value when the temperature increases beyond Tc . This tendency for the quasiparticle mass to decrease will be counterbalanced by the opposite tendency for the quasiparticle ‘thermal mass’ to increase with increasing temperature. As a result of these two counterbalancing tendencies in the

22 region of our interest, Tc < T < 2Tc , the effective mass of the light quarks are relatively constant. By examining the effects of the light quark quasiparticle masses on the quark-gluon plasma equation of state, Levai et al. [66], Szabo et al. [67], and Ivanov et al. [68] estimate that mq is about 0.3 to 0.4 GeV at Tc < T < 2Tc . As in the case of T = 0, where light quarks with a constituent-quark mass of about 350 MeV mimic the effects of chiral symmetry breaking and non-relativistic constituent quark models have been successfully used for light hadron spectroscopy [62, 63], so the large value of the estimated quasiparticle mass (from 0.3 to 0.4 GeV) may allow the use of a non-relativistic potential model as an effective tool to estimate the stability of light quarkonia at Tc < T < 2Tc . It will be of interest to investigate the relativistic effects [48, 49] in the future. For light quark masses of 0.3 to 0.4 GeV, we can estimate from the results in Fig. 13 for the W1 potential that as a quarkonium with light quarks has a reduced mass of 0.15-0.2 GeV, it may be bound at temperatures below (1.05 − 1.07)Tc. An open heavy quarkonium with a light quark and a heavy antiquark or a light antiquark and a heavy quark have a reduced mass of about 0.3-0.4 GeV and may be bound at temperatures below (1.11 − 1.19)Tc. Another lattice gauge calculation gives mq /T = 3.9 ± 0.2 at 1.5Tc [69], which implies that at T =1.5Tc (or about 0.3 GeV), the quark mass will be ∼1.2 GeV for (u, d, s) quarks. Such a ‘light’ quark quasiparticle mass appears to be quite large and may be uncertain, as the plasma will have approximately equal abundances of ‘light’ and charm quarks, which is however not observed. There may also be difficulties in reproducing the plasma equation of state. With this mass, a ‘light’ quarkonium will have a reduced mass of 0.6 GeV, and the quarkonium may be bound at temperatures below ∼1.31Tc. In either case, the drip lines of Fig. 13 for full QCD with 2 flavors obtained with the W1 potential do not support ¯ states with light quarks beyond 1.5Tc. A recent study of baryon-strangeness correlations suggests that the bound QQ ¯ component at 1.5Tc [44]. quark-gluon plasma contains essentially no bound QQ XII.

CONCLUSIONS AND DISCUSSIONS

The degree to which the constituents of a quark-gluon plasma (QGP) can combine to form composite entities is an important property of the plasma. To study the composite nature of the plasma, we need to examine the stability ¯ potential. We seek to extract the Q-Q ¯ potential of quarkonium in quark-gluon plasma which depends on the Q-Q from thermodynamical quantities obtained in lattice gauge calculations. For such a purpose, we need the relationship ¯ potential and the internal energy obtained in lattice gauge calculations. Such a relationship was between the Q-Q derived previously in Ref. [27]. We would like to gain additional support by examining whether a similar relationship ¯ potential and the internal energy in an analogous, but not identical, case of Debye screening. exists between the Q-Q ¯ under Debye screening, (1) the potential for the Q and Q ¯ in the We find that in adiabatic motion of Q and Q ¯ ¯ Schr¨odinger equation contains the interactions that act on Q and Q, (2) this Q-Q potential under Debye screening is only part of the total internal energy of the system, (3) the other part of the internal energy is the internal energy of the medium particles, and (4) many thermodynamical quantities such as the number, the entropy, and the internal ¯ in the grand canonical ensemble. energy of the medium particles increases with the separation between Q and Q Therefore, to obtain the Debye screening potential between two static charges, it is necessary to subtract out the internal energy of the medium particles from the total internal energy. These results supports a similar conclusion reached earlier in the analogous lattice gauge theory [27]. ¯ potential in the quark-gluon plasma by subtracting out the internal energy We are thus led to obtain the Q-Q of the medium particles from the total internal energy in the grand canonical ensemble. We proposed a method to subtracting out the internal energy of the medium by making use of the equation of state of the quark-gluon plasma obtained in an independent lattice gauge calculation [27]. The potential can then be represented as a linear combination of U1 and F1 , with coefficients depending on the quark-gluon plasma equation of state. The proposed potential in the quenched approximation is found to give dissociation temperatures that agree with those from spectral function analyses. It can be generalized to the case of full QCD to discuss quarkonium states in the plasma. The knowledge of the single-particle states using potentials extracted from lattice gauge calculations in full QCD can then be used to examine the limit of stability of both heavy and light quarkonia and to determine the location of the quark drip lines. The quark drip lines allows one to ascertain the degree of stability of heavy and light quarkonia when the masses of the quarks are known. J/ψ, χc , Υ, χb , and Υ′ are found to be stable in the plasma and dissociate at different temperatures. The characteristics of the quark drip lines severely limit the region of possible quarkonium states with light quarks to temperatures close to the phase transition temperature. Various estimates give a light quark mass of about 0.3-0.4 GeV [66, 67, 68], which is not very different from the constituent quark masses in non-relativistic quark models of hadrons. Bound quarkonia with light quarks may exist very near the phase transition temperature if their

23 effective quark mass is of the order of 300-400 MeV and higher. The author thanks Drs. H. Crater and Su-Houng Lee for helpful discussions. This research was supported in part by the Division of Nuclear Physics, U.S. Department of Energy, under Contract No. DE-AC05-00OR22725, managed by UT-Battle, LLC and by the National Science Foundation under contract NSF-Phy-0244786 at the University of Tennessee.

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